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% 信息设置
\title[ch10]{Chapter 10: Holonomic D-modules}
\author[]{SCC LQW}
%\institute[XX大学]{XX大学\quad 数学与统计学院\quad 数学与应用数学专业}
%\date{2025年6月}

\begin{document}

%---------------------------------------------------
% 封面页
\begin{frame}
  \titlepage
\end{frame}

%---------------------------------------------------
% 目录页
\begin{frame}{Contents}
  \tableofcontents
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 1
\section{DEFINITION AND EXAMPLES.}

%---------------------------------------------------
\begin{frame}{1.0 DEFINITION.}

%\begin{mydefinition}
Let $n$ be a positive integer.
A finitely generated left $A_n$-module is holonomic if it is zero, or if it has dimension $n$. 
Recall that by Bernstein's inequality this is the minimal possible dimension for a non-zero $A_n$-module.
%\end{mydefinition}

\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item The dimension of a $D$-module is defined by the degree of the Hilbert polynomial of the associated graded module over the symmetry algebra $S_{2n}$. 
\item 
\item 
\end{itemize}

\end{frame}

%---------------------------------------------------
\begin{frame}{1.1 PROPOSITION. }
%\begin{myproposition}
(1) Submodules and quotients of holonomic $A_n$-modules are holonomic. 

(2) Finite sums of holonomic $A_n$-modules are holonomic.
%\end{myproposition}

\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item 
\item 
\item 
\end{itemize}

\end{frame}

%---------------------------------------------------
\begin{frame}{1.2 COROLLARY.}
%\begin{mycorollary}
Finitely generated torsion $A_1$-modules are holonomic.
%\end{mycorollary}

\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item 
Let $P\in A_1$ and $P\neq 0$, then $M=A_1/A_1P$ is a holonomic $A_1$-module. 
\item 
\item 
\end{itemize}

\end{frame}

%---------------------------------------------------
\begin{frame}{1.3 PROPOSITION.}
%\begin{myproposition}
Holonomic $A_n$-modules are torsion modules.
%\end{myproposition}

\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item A torsion module is a module $M$ that each element is torsion, i.e., annihilated by some non-zero element in $A_n$. 
\item If $m\in M$ is not torsion, then $A_nm$ is isomorphic to $A_n$ as left $A_n$-modules. 
\item 
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 2
\section{BASIC PROPERTIES.}

%---------------------------------------------------
\begin{frame}{2.1 THEOREM. (artinian modules)}
%\begin{mytheorem}
Let $M$ be a left module over a ring $R$ and let $N$ be a submodule of $M$.

(1) $M$ is artinian if and only if every set $\mathcal{S}$ of submodules of $M$ has an element (a minimal element) which does not contain any other element of $\mathcal{S}$.

(2) $M$ is artinian if and only if $N$ and $M/N$ are artinian.

(3) Let $N{\,}'$ be another submodule of $M$, and suppose that $M = N + N{\,}'$. 
If $N, N{\,}'$ are artinian, then so is $M$.
%\end{mytheorem}

\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item 
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\end{itemize}


\end{frame}

%---------------------------------------------------
\begin{frame}{2.2 THEOREM.}
%\begin{mytheorem}
Holonomic modules are artinian.
%\end{mytheorem}

\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item 
\item 
\item 
\end{itemize}


\end{frame}

%---------------------------------------------------
\begin{frame}{2.3 SCHOLIUM. (A scholiast makes a scholium. ) }
%\begin{myscholium}
The length of a holonomic module cannot exceed its multiplicity.
%\end{myscholium}

\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item 
\item 
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\end{itemize}

\end{frame}

%---------------------------------------------------
\begin{frame}{2.4 COROLLARY. }
%\begin{mycorollary}
A holonomic $A_n$-module of multiplicity 1 is irreducible.
%\end{mycorollary}

\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item 
\item 
\item 
\end{itemize}

\end{frame}

%---------------------------------------------------
\begin{frame}{2.5 THEOREM.}
%\begin{mytheorem}
Let $R$ be a simple left noetherian ring and $M$ a finitely generated left $R$-module. 

If $M$ is artinian but $R$ is not artinian (as a left $R$-module), then $M$ is a cyclic module.
%\end{mytheorem}
\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item Example. {\color{red}($Z$ is not a simple ring.)} The $\mathbb{Z}$-module $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$ is not cyclic. 
\item Example. {\color{red}($A_1$ is a simple ring.)} Let $P,Q\in A_1$ be nonzero elements.
By this theorem, the $A_1$-module $A_1/A_1P\oplus A_1/A_1Q$ is cyclic, 
i.e., there exists an element $R\in A_1$ such that $$A_1/A_1P\oplus A_1/A_1Q\cong A_1/A_1R. $$
How do you construct $R$ from $P$ and $Q$? 

\item 
\end{itemize}

\end{frame}

%---------------------------------------------------
\begin{frame}{2.6 COROLLARY. }
%\begin{mycorollary}
Holonomic modules are cyclic.
%\end{mycorollary}
\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item 
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\end{itemize}

\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 3
\section{FURTHER EXAMPLES.}

%---------------------------------------------------
\begin{frame}{3.1 LEMMA.}
%\begin{mylemma}

Let $M$ be a left $A_n$-module. 

Let $\Gamma$ be a filtration of $M$ with respect to the Bernstein filtration of $A_n$. 

Suppose that there exist constants $c_1, c_2$ such that for $t \gg 0$

$$
\dim_K \Gamma_t \leq \frac{c_1}{n!} t{\,}^n + c_2 t{\,}^{n-1}.
$$

Then $M$ is a holonomic $A_n$-module whose multiplicity cannot exceed $c_1$. 

In particular $M$ is finitely generated.

%\end{mylemma}

\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item 
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\end{itemize}

\end{frame}

%---------------------------------------------------
\begin{frame}{3.2 THEOREM. }
%\begin{mytheorem}
The $A_n$-module $K[X, p^{-1}]$ is holonomic and its multiplicity is $\leq (\deg(p) + 1)^n$.
%\end{mytheorem}

\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item 
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\end{itemize}

\end{frame}

%---------------------------------------------------
\begin{frame}{3.3 THEOREM.}
%\begin{mytheorem}
Let $p \in K[X]$. There exist a polynomial $B(s) \in K[s]$ and a differential operator $D(s)$ in the polynomial ring $A_n(K)[s]$ such that
$$
B(s) p^s = D(s) p p^s.
$$
%\end{mytheorem}
\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item 
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\end{itemize}

\end{frame}

%---------------------------------------------------
\begin{frame}{3.4 DEFINITION. }
%\begin{mydefinition}
The polynomial $B(s)$ and the operator $D(s)$ are not uniquely determined by $p$. 

However, the set of all the possible polynomials $B(s)$ satisfying Theorem 3.3 is an ideal of $K[s]$, as one can easily verify. 

The monic generator of this ideal is thus unique; it is called the Bernstein polynomial of $p$ and denoted by $b_p(s)$.
%\end{mydefinition}

\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
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\end{itemize}

\end{frame}

%---------------------------------------------------
\begin{frame}{3.5 EXAMPLE.}
%\begin{myexample}
Let $p = x_1^2 + \cdots + x_n^2$. 

If we denote by $D$ the differential operator $\partial_1^2 + \cdots + \partial_n^2$ then

$$
D \cdot p^{s+1} = 4(s+1)(s+n/2) p^s.
$$

Thus $b_p(s) = 4(s+1)(s+n/2)$. 

Note that in this case $D$ is independent of $s$.
%\end{myexample}

\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item 
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\end{itemize}

\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 4
\section{EXERCISES.}

%---------------------------------------------------
\begin{frame}{EXERCISE 1}
% E1 

Show that $A_n / A_n \partial_n$ is a torsion $A_n$-module.

Hint: Every class in $A_n / A_n \partial_n$ contains a representative of the form 

$$\sum_{i=0}^{r} d_i x_n^i$$ 

where $d_i \in A_{n-1}$. 

Show that this element is annihilated by $\partial_n^{r+1} \in A_n / A_n \partial_n$.

\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item 
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\end{itemize}

\end{frame}

%---------------------------------------------------
\begin{frame}{EXERCISE 2}
% E2 

Let $I$ be a non-zero left ideal of $A_n$. Show that $I$ is not an artinian $A_n$-module.

\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item 
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\end{itemize}


\end{frame}

%---------------------------------------------------
\begin{frame}{EXERCISE 3}
% E3 

Show that $A_n / A_n \partial_n$ is not an artinian $A_n$-module.

Hint: The modules

$$
(A_n x_n^k + A_n \partial_n) / A_n \partial_n
$$

form an infinite descending chain in $A_n / A_n \partial_n$.

\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item 
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\end{itemize}



\end{frame}


%---------------------------------------------------
\begin{frame}{EXERCISE 4}
% E4 

Let $p \in K[X]$ be a non-constant polynomial. Is the module $K[X, p^{-1}]$ irreducible?


\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
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\end{itemize}



\end{frame}


%---------------------------------------------------
\begin{frame}{EXERCISE 5}
% E5 

For $r \leq n$, let 

$$
P = \sum_{i=1}^{r} \partial_i^2 - \sum_{i=r+1}^{n} \partial_i^2
$$ 

Find a left ideal $J$ of $A_n$ such that $A_n P \subseteq J$ and $A_n / J$ is holonomic.


\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item 
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\end{itemize}



\end{frame}


%---------------------------------------------------
\begin{frame}{EXERCISE 6}
% E6 

Let $p \in K[X]$. Show that 

$$K(s)[X, p^{-1}] p^s$$ 

is a holonomic $A_n(K(s))$-module.

Hint: Let $m$ be the degree of $p$. The $K$-vector spaces, 

$$\Gamma_k = \{ q \cdot p^{-k} \cdot p^s : \deg(q) \leq (m+1)k \}$$ 

give rise to a filtration of $K(s)[X, p^{-1}]$. What are their dimensions?

\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item 
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\end{itemize}



\end{frame}


%---------------------------------------------------
\begin{frame}{EXERCISE 7}
% E7 

Let $p$ be a polynomial in $K[x_1, \ldots, x_n]$. 

Denote by $A_n[s] \cdot p^s$ the submodule of $K[s, X, p^{-1}]$ generated by $p^s$ over the polynomial ring $A_n[s]$. 

This is also an $A_n$-module, but in this case it is not clear whether it is finitely generated! 

Show that if $p$ belongs to the ideal of $K[X]$ generated by its partial derivatives, then $A_n[s] \cdot p^s$ is finitely generated as an $A_n$-module.

Hint: Let 

$$
D = \sum_{i=1}^{n} \frac{\partial p}{\partial x_i} \partial_i
$$ 

Then $D \cdot f = s f$.

\end{frame}

%---------------------------------------------------
\begin{frame}{EXERCISE 7 Solution}
% E7 

\begin{itemize}
\item 
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\end{itemize}



\end{frame}

%---------------------------------------------------
\begin{frame}{EXERCISE 8}
% E8 

Let $p$ be a polynomial in $K[x_1, \ldots, x_n]$. 
Let  $$M = A_n[s] \cdot p^s$$ 

be the submodule of $K[s, X, p^{-1}]$ generated by $p^s$ over the polynomial ring $A_n[s]$. 

Let $t : M \to M$ be the map defined by 

$$t(D(s) \cdot p^s) = D(s+1) p \cdot p^s$$

(1) Show that $t$ is an endomorphism of $M$ as an $A_n$-module but \underline{not} as an $A_n[s]$-module.

(2) Show that $[t, s] = t$.

(3) Use (2) to show that $M / tM$ is an $A_n[s]$-module, even though $t$ is not $A_n[s]$-linear.

(4) Show that $M / tM$ is a holonomic module. 

\end{frame}

%---------------------------------------------------
\begin{frame}{EXERCISE 8 Solution}
% E7 

\begin{itemize}
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\end{itemize}


\end{frame}


%---------------------------------------------------
\begin{frame}{EXERCISE 9}
% E9 

Let $\lambda_1, \ldots, \lambda_n$ be real numbers. 
%
Set 
$$f(x_1, \ldots, x_n) = \exp(\lambda_1 x_1 + \cdots + \lambda_n x_n)$$

(1) Find generators for the left ideal 

$$J = \{ P \in A_n(\mathbb{R}) : P \cdot f = 0 \}$$

(2) Show that $A_n(\mathbb{R}) / J$ is a holonomic module over $A_n(\mathbb{R})$.

\noindent\rule{\textwidth}{0.4pt}

\begin{itemize}
\item 
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\end{itemize}


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}

